Volume 27, Number 5 November 1996 

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The theme "Building Bridges of Understanding" is especially important to the mathematics education community. It proclaims the next steps that we all must take to make implementing the NCTM's Standards a longlasting, successful effort to enhance mathematics education. Over the past two years, I have seen exciting changes taking place in many classrooms:
As we move forward, charged with making the ideals promoted in the NCTM's Standards publications a commonplace reality, we are met by myriad groups that either share in our efforts or scorn our strategies. We are propelled by strong support, yet we are faced with firm opposition.
It is important to note that we are all working toward one common goal: improving mathematics education for all. To succeed, we must educate our children better, prepare our teachers better, and inform our parents better. We must build bridges of understanding for all.
BUILDING BRIDGES TO OUR CHILDREN
Many of us can easily remember mathematics's being taught through rote and drill when we memorized multiplication tables, simplified algebraic expressions without end, and completed rows and rows of longdivision problems using DMSB (divide, multiply, subtract, and bring down). What we now know, looking back, is that mathematics teaching fell short when it came, for example, to students' understanding of when and where mathematics could be applied. We may have had the procedural knowledge, but rarely did we have the conceptual knowledge. We were told, "To divide by a fraction, invert the divisor and multiply." Right! Which one is the divisor?
Do we then advocate that no procedural knowledge be taught? Of course not. Algorithms and strong basic arithmetic skills are as important as they ever were. But we believe that students also need to understand.
The research on nonsense problems only bears out what we have suspected, namely, that students faced with a problem that does not make sense begin to solve it anyway because, after all, mathematics does not necessarily make sense to them. For example: "A shepherd has 125 sheep and 5 dogs. How old is the shepherd?" Or consider a hotel guest in an elevator in Las Vegas who pushed the buttons for the tenth and sixth floors, expecting to get to the sixteenth floor. The mathematics is right, but it does not get you where you want to go.
By connecting mathematics to the real world, students are able to apply the mathematics that they learn in the classroom to reallife situations. By connecting mathematics to other disciplines, students can see how much the world relies on mathematics. They are also able to connect pieces of mathematics that have often been taught in separate compartments. These fundamental assumptions underlie the NCTM's Standards. The Standards seek to change how students are taught mathematics so that they will, in turn, be provided a strong mathematics foundation that will enable them to meet the challenges of their future. To make the changes a reality, the following must occur:
How refreshing it is to hear from students who have had various positive mathematics experiences. Alex Hryhorczuk, a sixth grader at Roosevelt Middle School in River Forest, Illinois, believes that the elementarylevel University of Chicago School Mathematics Project is "fun, interesting, and teaches a lot of mathematics." According to Alex, "You constantly face new problems instead of the same type over and over again. This way you learn more in a shorter amount of time and become a quicker thinker."
It is compelling to know that Julia Rankin of Hogan High School in Vallejo, California, compares the Applications Reform in Secondary Education (ARISE) project with a "reality check." Julia states that the project "has done for me what no other mathematics classes did. It has given me the skills I will use in years to come."
It is exciting to hear Kelsey Robinson, a ninth grader at the School for the Arts in San Francisco, describe her experiences in the Middle School Mathematics Project. She was taught mathematics through reallife situations, such as designing a house, and she realized that it "will help me when I design a stage set or a laundry room. How often, when someone has a problem, do they think back and say, 'Oh, I can use what I learned in middle school to solve my problem'?"
These students credit today's mathematics education with helping them become better mathematics students and more mathematically skilled citizens.
Perhaps Luis Cruz, a junior at the University of California at Berkeley, sums it up best as he reflects on his high school mathematics experience. Luis credits the Interactive Mathematics Program (IMP), which encourages students to communicate mathematically and includes class presentations, with helping him overcome his fear of public speaking. He states, "Probably the best thing I got out of IMP, in addition to math concepts, was confidence. Now, I am no longer afraid to try new things. I am confident that I can perform any task assigned to me."
I am sure that each of you has comments such as these that you can share with your colleaguescomments that give us hope, that inspire us, and that propel us forward, undaunted.
Our students are our greatest resource. They are the positive examples that we must hold up as proof that the teaching and learning of mathematicsspanning prekindergarten through graduate schoolis improving.
BUILDING BRIDGES TO OUR TEACHERS
Fields Medal winner William Thurston noted in the January 1990 issue of Quantum that
[w]ith the modern emphasis on test scores, on "basics," on mathematics as a competitive sport, on getting "ahead" in math, and so on, it often seems that the diversity, richness, liveliness, and depth of mathematics has been pruned away from the school experience. Mathematics isn't a palm tree, with a single long straight trunk covered with scratchy formulas. It's a banyan tree that has grown to the size of a forest, inviting us to climb and explore.
We are fortunate to have innovative teachers who have taken it on themselves to plant banyan trees in classrooms nationwide.
For our part, NCTM has worked diligently to ensure that mathematics teachers are given adequate professionaldevelopment opportunities. Journals; books; annual, regional, and special meetings; and now a Web page have built bridges to allow even the most isolated teacher to stay uptodate. Furthermore, we have increased the opportunities for teachers to participate in the governance and activities of the Council.
BUILDING BRIDGES TO OUR PARENTS
What has been most distressing since we released the Standards documents is that our efforts to inform parents better have fallen short.
Throughout the past year, phrases like fuzzy math and new new math have been coined to describe the NCTM's Standards. Some of our opponents claim that the answer to what plagues mathematics education is a prompt return to the basics. You may recall that a return to the basics is what destroyed the last attempt to improve mathematics education and led to a dramatic downturn in test scores.
The idea of initiating a backtobasics movement implies that we have abandoned the basics, which simply is not true. The strength of the Standards rests in their call for the continuation of the "basics" as most people know themaddition, subtraction, multiplication, and divisionas well as the introduction of the businessandindustry "basics," which include reasoning, problem solving, communicating, making mathematical connections, and collaborating.
We do teach the basicsthe basics of today and tomorrow. In the film Apollo 13, a problem arose on the spacecraft. The astronauts were rapidly losing their oxygen, so groundcrew members gathered all the equipment that they knew was on the capsule, took it to a group of engineers, dropped it on the table, and said, "This is the problem. This is what you have to work with. Solve the problem." You did not see an engineer grab a piece of equipment and run off alone to a corner. You saw people working together to solve a problem.
There is nothing "fuzzy" about the mathematics promoted in the Standards documents. In fact, the Standards promote higher levels of mathematics earlier for all students. Some parents are concerned that because other, previously disenfranchised students can now accomplish mathematics, this mathematics is not good enough for their children. This attitude has developed in spite of their mathematically promising students' having said in interviews that they have never been more challenged by, or more interested in, mathematics. We have to help those parents bridge their fears and encourage them to join hands in providing a solid mathematics education for all children.
BUILDING BRIDGES TO OUR COLLEAGUES
The changes in content and pedagogy proposed by the Standards have bothered some of our college colleagues. If the NCTM's Standards are right, are the content and teaching practices to which they have devoted their lives wrong? Of course not. That content and those teaching practices were acceptable when schools were expected to graduate a much lower percentage of the student body. They are not acceptable for the technological world of the twentyfirst century and the diversity that we find in today's classrooms. All young people deserve a significant mathematics education, made possible by the use of teaching strategies that are congruent with their learning styles.
We have learned a great deal during the past twenty years about how children learn and how individual learning styles differ. We need to use the results of that research. To use only one method to try to reach all the students at whatever levels is negligent at best and criminal in the extreme.
A few weeks ago, I could not carry out a function on my computer because I did not have enough remaining memory. I was told to eliminate some of the material on the disk to clear some memory. While scrolling through the menu, I came across files on Math 205, 206, and 207 and TED 425the courses that I normally teach at California State Polytechnic University. I thought, "Wait a minute. If I truly believe that every class is different from every other class and that every student is different from every other student, then that stuff's got to go." And so I deleted syllabi, class schedules, lesson plans, and tests. We all need to make these same kinds of decisions about what we really believe. We need to build bridges to our colleagues to help them broaden their horizons to help more students.
BUILDING BRIDGES TO OUR FUTURE
We no longer have to ask, "Are the Standards working?" We now have evidence that they are.
But as we look forward, building bridges to our future, we have to ask ourselves, "Are the Standards working for every child? Are we, in fact, providing an equal opportunity for all students to learn mathematics?"
When I took office two years ago, I made equity a top priority. Equity does not mean treating everybody equally. Just go to a football stadium or theater containing an equal number of men's rooms and women's rooms. Is that an equitable situation? The numbers may be equal, but that does not take into consideration the differences in people that make it inequitable.
As classroom teachers, we see diversity every day when we look into the faces of our students. And that diversity reflects the workplace and society.
John F. Kennedy, a staunch supporter of diversity, once said the following:
Let us not be blind to our differencesbut let us also direct attention to our common interests and the means by which those differences can be resolved. And, if we cannot end now our differences, at least we can help make the world safe for diversity.
Today, I see vast improvements in reaching all students. We have strengthened our efforts to teach students, both female and male, of various cultures and backgrounds. Our San Diego location offers a metaphor: Here on the water, a rising tide lifts all boats. Higher expectations lift all children.
Not every student will grow up to become a mathematician. But every student can learn mathematics. It is our responsibility to give every student opportunities to learn as well as opportunities to express her or his mathematical skills.
As I conclude my twoyear term as president, I leave you with the words of Constance Clayton, former president of the Council of the Great City Schools, who summed up our challenge most eloquently in these words:
If we can bail out our savings and loans, we can lift up our children. If we can build more prisons, we can keep our schools from looking like them. If we can fashion the weapons of war, we can mastermind the tools for learning.
NCTM has taken great strides in developing the tools needed to improve mathematics education. And we have excelled in our efforts to give today's students the mathematics opportunities to make them tomorrow's successful citizens.
As we head into the turn of the century, I charge each and every one of you, from the prekindergarten through the graduatelevel educator, to stand firm in your beliefs that every student can learn mathematics, that every teacher must have adequate support and professionaldevelopment opportunities, and that every parent has a vested interest in our common goals of setting and achieving higher standards for mathematics education.
To paraphrase another great person, Dr. Martin Luther King, Jr., I have a dream that working together we can build bridges of understanding that will stand the test of time and that will carry every child forward to achieve excellence in mathematics.